d z / d t = - (1 + z) H

\frac{d l o g ρ _{DE}}{d z} = \frac{3 [ 1 + w ( z )]}{1 + z} .

ρ_{DE} (z) = ρ_{DE, 0} (1 + z)^{3} exp [3 \int_{0}^{z} \frac{w ( z ^{'} )}{1 + z ^{'}} d z^{'}] .

Combining this with

ρ_{m} = ρ_{m, 0} (1 + z)^{3}

H (z) = H_{0} {Ω_{m, 0} (1 + z)^{3} + Ω_{DE, 0} (1 + z)^{3} exp [\int_{0}^{z} \frac{3 w ( z ^{'} )}{X ( z ^{'} )} d z^{'}]}^{1/2},

where

X (z) = 1 + z .

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